Hybrid metal-semiconductor heterostructures represent tunable photonic materials with potential applications in coherent light sources. In such heterostructures, cooperative dynamics of excitons and photon-emission properties are strongly influenced by collective surface plasmon modes of metal nanoparticles constituting a plasmonic cavity. To gain insight into the cooperative dynamics of quantum emitters in such a cavity, we have developed a quantum mechanical approach treating the quantum emitter, surface plasmon, and photon decrees of freedom on the same footing. This allows us to account for multiple-scale interactions between plasmon modes and quantum emitters. Furthermore, the interactions can naturally be partitioned into coherent and incoherent couplings resulting in a generalized version of plasmonic Dicke model. Examination of associated phase diagram in both equilibrium and non-equilibrium regimes allows us to identify the polariton condensate, superradiant, and various lasing regimes. Finally, we performed numerical simulations of plasmonic response of a linear array formed by either metal nanorods or asymmetric metal nanoantennas. The calculations show that such nanostructures have potential for achieving strong coupling regime and realizing predicted cooperative regimes.